Power Reduction Formulas/Hyperbolic Cosine Squared

Theorem

$\cosh^2 x = \dfrac {\cosh 2 x + 1} 2$

where $\cosh$ denotes hyperbolic cosine.

Proof 1

 $\displaystyle 2 \cosh^2 x - 1$ $=$ $\displaystyle \cosh 2 x$ Double Angle Formula for Hyperbolic Cosine:Corollary 1 $\displaystyle \cosh^2 x$ $=$ $\displaystyle \frac {\cosh 2 x + 1} 2$ solving for $\cosh^2 x$

$\blacksquare$

Proof 2

 $\displaystyle \cosh^2 x$ $=$ $\displaystyle \frac 1 4 \paren {e^x + e^{-x} }^2$ Definition of Hyperbolic Cosine $\displaystyle$ $=$ $\displaystyle \frac {e^{2 x} + e^{-2 x} + 2} 4$ $\displaystyle$ $=$ $\displaystyle \frac {\cosh 2 x + 1} 2$ Definition of Hyperbolic Cosine

$\blacksquare$

Proof 3

 $\displaystyle \cosh^2 x$ $=$ $\displaystyle \cos^2 i x$ Hyperbolic Cosine in terms of Cosine $\displaystyle$ $=$ $\displaystyle \frac {\map \cos {2 i x} + 1} 2$ Square of Cosine $\displaystyle$ $=$ $\displaystyle \frac {\cosh 2 x + 1} 2$ Hyperbolic Cosine in terms of Cosine

$\blacksquare$